Linear Equations in Two
Variables
Graphing Linear Equations
Writing Equations and Graphing
• These activities introduce rates of change and
defines slope of a line as the ratio of the vertical
change to the horizontal change.
• This leads to graphing a linear equation and
writing the equation of a line in three different
forms.
The Coordinate Plane
Graphing ordered pairs in the
coordinate plane
Coordinate
axes
origin
quadrants
ordered
pair
The coordinate plane is formed by
placing two number lines called
coordinate axes so that one is
horizontal (x-axis) and one is
vertical (y-axis). These axes
intersect at a point called the
origin (0, 0), which is labeled 0.
An ordered pair (x, y) is a pair of
real numbers that correspond to a
point in the coordinate plane. The
first number in an ordered pair is
the x-coordinate and the second
number is the y-coordinate.
In a previous lesson, you
learned to plot points from
a table of values. The
solution to an equation in
two variables is all ordered
pairs of real numbers (x, y)
that satisfy the equation.
You can graph the solution
to an equation in two
variables on the coordinate
plane by using the (x, y)
values from the table.
Graph each ordered pair
in the table. Do all the
points appear to lie on the
line?
Make a table of ordered pairs for each
equation. Then graph the ordered
pairs. (x = -3, -2, -1, 0, 1, 2, 3)
1) y = x + 3
2) y = x – 2
3) y = 2x
4) y = -2x
5) y = - 2
6) y = -x + 1
7) y = x2 – 1
8) y = 3
9) y = 2x2 - 1
In everyday life, one
quantity often depends on
another. On an automobile
trip, the amount of time that
you spend driving depends on
your speed. In cases like
these, it is said that the
second quantity is a function
of the first. The second
quantity is called dependent,
while the first is
independent.
Can you think of other
situations in real-life where
one quantity is dependent on
another.
Many functions (equation)
can be represented by an
equation in two variables.
The variable that
represents the dependent
quantity is called the
dependent variable. The
variable that represents
the independent quantity
is called the independent
variable. When you write
an ordered pair using
these variables (x, y), the
independent variable
appears first.
(independent, dependent)
In the function y = 2x, each
x-value is paired with
exactly one y-value. Thus
the value of y depends on
the value of x.
In a function, the variable of
the domain is called the
independent variable and
the variable of the range is
called the dependent
variable. On a graph, the
independent variable is
represented on the
horizontal axis and the
dependent variable is
represented on the vertical
axis.
Example
The value v (in cents) of n
nickels is given by the
function (equation) v = 5n
a) Identify the
independent and dependent
variable.
b) Represent this function
in a table for n = 0, 1, 2,
3, 4, 5, 6.
c) Represent this function
in a graph.
Domain
Range
Use both a table and a graph to
represent each function for the
given values of the independent
variable.
1) The value v of 0, 1, 2, 3, 4, 5, and 6
dimes is a function of the number of
dimes, d.
2) The distance d traveled by a
jogger who jogs 0.1 miles per minute
over 0, 1, 2, 3, 4, and 5 minutes is a
function of time, t.
Try another
3) One hat cost $24. The total cost c
of 1, 2, 3, 4, 5, and 6 hats is a function
of the number n of hats purchased.
A 1200-gallon tank is empty. A valve is
opened and water flows into the tank at the
constant rate of 25 gallons per minute.
1) How would you find the amount of water in
the tank after 5 minutes, 6 minutes, and 7
minutes?
2) How would you find the time it takes for
the tank to contain 500 gallons, 600 gallons,
and 700 gallons?
What is the independent variable?
What is the dependent variable?
What function represents this situation?
Slope and Rate of
Change
Finding the slope of a Line
Slope – A measure of the steepness of a line on a
graph; rise divided by the run.
If P (x1, y1) and Q (x2, y2) lie along a nonvertical
line in the coordinate plane, the line has slope m,
given by
Linear equation – An equation whose solutions
form a straight line on a coordinate plane.
Collinear – Points that lie on the same line.
Remember, linear equations have constant
slope. For a line on the coordinate plane, slope
is the following ratio. This ratio is often
referred to as “rise over run”.
Slope and
Orientation of a Line
In the coordinate
plane, a line with
positive slope rises
from left to right. A
line with negative
slope falls from left
to right. A line with 0
slope is horizontal.
The slope of a
vertical line is
undefined.
Find the slope of the line that passes through
each pair of points.
1)
(1, 3) and (2, 4)
2)
(0, 0) and (6, -3)
3)
(2, -5) and (1, -2)
4)
(3, 1) and (0, 3)
5)
(-2, -8) and (1, 4)
Finding points on a Line
Graphing a line using a point
and the slope is shown here.
Graph the line passing
through (1, 3) with slope 2.
a) Graph point (1, 3).
b) Because the slope is 2,
you can find a second point
on the line by counting up 2
units and then right 1 unit.
C) Draw a line through point
(1, 3) and the new point.
A line with slope –¼
contains P(0, 5).
a) Sketch the line
b) Find the coordinates
of a second point on the
line.
Sketch each line from the information given.
Find a second point on the line.
a) point H(0, 0), slope 2
b) point S(3, 3), slope -3
c) point J( 2, 5); slope –1/2
d) point T(-3, 1); slope – 2/5
Finding Slope from a
Graph
Choose two points on
the line (-4, 4) and (8, 2). Count the rise over
run or you can use the
slope formula. Notice if
you switch (x1, y1) and (x2,
y2), you get the same
slope:
Real-world Applications
You have been solving problems
in which quantities change since
early school days. For instance,
here is a typical first-grade
problem:
If I have two apples, and
then I buy five more apples,
how many apples do I have in
all?
This situation concerns a change
in the number of apples.
Try this problem
One marker on a road showed 20
miles. The next marker showed
170 miles. How many miles have
gone by?
The time was 2:00 pm at the first
marker. Then it was 5:00 pm at
the second marker. How may
hours have passed?
150/3 compares
quantities and is
called rate.
As you can see, simple subtraction
is all that is required to answer
the questions.
The answers, 150 miles and 3
hours, indicate a change in miles
and a change in time, respectively.
Since each quantity describes a
change, the expression is a rate of
change. You read the rate as
150 miles in 3 hours
or
150 miles per 3 hours.
Rates can be positive (+)
or
negative (-)
In everyday life, most rates are
given “per one unit” of a quantity.
This is called a unit rate. So, it is
more common to perform the
division 150/3 = 50 to arrive at a
unit rate of change:
50 miles per 1 hour
or
50 miles per hour.
After stopping to buy gas,
a motorist drives at a
constant rate as indicated
below.
Time (t) 0 hrs
Dist (d) 5 mi
2 hrs
120 mi
Find the speed or rate of
change for the car then
graph the situation.
Problem solving
Calculate each rate of
change
1) A motorist drives 140
miles in 3 hours and 280
miles in 4.4 hours.
2) After 5 minutes a
temperature of 35° F was
recorded. After 6.5
minutes, the temperature
was 32° F.
3) A hose flows 250
gallons of water in 2
minutes and 875 gallons in
7 minutes.
4) A small aircraft begins a descent to land. At 1.5
minutes the altitude is 5450 feet, and at 3 minutes the
altitude is 4700 feet. If the plane descends at a
constant rate, what is the rate of descent.
5) 32 people exit the stadium in 2.5 minutes and 384
people leave the stadium in 30 minutes.
6) A race car drives 2 miles in 0.02 hours and 305
miles in 3.02 hours.
7) After 3.5 minutes of descent, a small plane’s
altitude is 3425 feet. After 6.0 minutes, the plane’s
altitude if 2375.
8) Does the following data indicate a constant speed?
Explain.
2 mi/4 min 8 mi/16 min 10 mi/24 min
A linear equation is an
equation whose
solutions fall on a line
on the coordinate
plane. All solutions of
a particular linear
equation fall on the
line, and all the points
on the line are
solutions of the
equation.
Look at the graph to
the left, points (1, 3)
and (-3, -5) are found
on the line and are
solutions to the
equation.
If an equation is linear, a
constant change in the xvalue produces a
constant change in the yvalue.
The graph to the right
shows an example where
each time the x-value
increases by 2, the
y-value increases by 3.
The equation
y = 2x + 6
is a linear equation
because it is the
graph of a straight
line and each time x
increases by 1 unit, y
increases by 2
X
Y = 2x + 6
Y
(x, y)
1
2
3
4
5
2(1) + 6
2(2) + 6
2(3) + 6
2(4) + 6
2(5) + 6
8
10
12
14
16
(1, 8)
(2, 10)
(3, 12)
(4, 14)
(5, 16)
Using Slopes and
Intercepts
x-intercepts and y-intercepts
x-intercept – the x-coordinate of the point
where the graph of a line crosses the x-axis
(where y = 0).
y-intercept – the y-coordinate of the point
where the graph of a line crosses the y-axis
(where x = 0).
Slope-intercept form (of an equation) – a linear
equation written in the form y = mx +b, where m
represents slope and b represents the yintercept.
Standard form (of an equation) – an equation
written in the form of Ax + By = C, where A, B,
and C are real numbers, and A and B are both ≠ 0.
Standard Form of an Equation
• The standard form of
a linear equation, you
can use the x- and yintercepts to make a
graph.
• The x-intercept is the
x-value of the point
where the line
crosses.
• The y-intercept is the
y-value of the point
where the line
crosses.
Ax + By = C
To graph a linear equation in standard form,
you fine the x-intercept by substituting 0 for
y and solving for x. Then substitute 0 for x
and solve for y.
2x + 3y = 6
2x + 3(0) = 6
2x = 6
x=3
2x + 3y = 6
2(0) + 3y = 6
3y = 6
y=2
The x-intercept is 3.
(y = 0)
The y-intercept is 2.
(x = 0)
Find the x-intercept and y-intercept of each
line. Use the intercepts to graph the equation.
1)
x–y=5
2)
2x + 3y = 12
3)
4x = 12 + 3y
4)
2x + y = 7
5) 2y = 20 – 4x
Slope-intercept Form
y = mx + b
Slope-intercept Form
• An equation whose
graph is a straight line
is a linear equation.
Since a function rule
is an equation, a
function can also be
linear.
• m = slope
• b = y-intercept
Y = mx + b
(if you know the slope and where
the line crosses the y-axis,
use this form)
For example in the equation;
y = 3x + 6
m = 3, so the slope is 3
b = +6, so the y-intercept is +6
Let’s look at another:
y = 4/5x -7
m = 4/5, so the slope is 4/5
b = -7, so the y-intercept is -7
Please note that in the slope-intercept formula;
y = mx + b
the “y” term is all by itself on the left side of the
equation.
That is very important!
WHY?
If the “y” is not all by itself, then we must first
use the rules of algebra to isolate the “y” term.
For example in the equation:
2y = 8x + 10
You will notice that in order to get “y” all by itself
we have to divide both sides by 2.
After you have done that, the equation becomes:
Y = 4x + 5
Only then can we determine the slope (4), and the yintercept (+5)
OK…getting back to the lesson…
Your job is to write the equation of a line
after you are given the slope and y-intercept…
Let’s try one…
Given “m” (the slope remember!) = 2
And “b” (the y-intercept) = +9
All you have to do is plug those values into
y = mx + b
The equation becomes…
y = 2x + 9
Let’s do a couple more to make sure you are
expert at this.
Given m = 2/3, b = -12,
Write the equation of a line in slope-intercept form.
Y = mx + b
Y = 2/3x – 12
One last example…
Given m = -5, b = -1
Write the equation of a line in slope-intercept form.
Y = mx + b
Y = -5x - 1
Writing an Equation From a Graph
You can write an equation
from a graph. Use 2 points
to find the slope. Then use
the slope and the yintercept to write the
equation.
Step 1 Using the slope formula,
find the slope. Two points on the
line are (0, 2) and (4, -1).
Step 2 write an equation in slopeintercept form. The y-intercept
is 2.
Write an equation of each line.
Use points (0, 1)
and (-2, 0)
Use points (0, 1)
and (3, -1)
Given the slope and y-intercept, write the
equation of a line in slope-intercept form.
1) m = 3, b = -14
2) m = -½, b = 4
3) m = -3, b = -7
4) m = 1/2 , b = 0
5) m = 2, b = 4
Using slope-intercept form
to find slopes and
y-intercepts
The graph at the right
shows the equation of a
line both in standard form
and slope-intercept form.
You must rewrite the
equation 6x – 3y = 12 in
slope-intercept to be able
to identify the slope and yintercept.
Using slope-intercept form to write equations,
Rewrite the equation solving for y = to
determine the slope and y-intercept.
3x – y = 14
-y = -3x + 14
-1 -1 -1
y = 3x – 14
or
3x – y = 14
3x = y + 14
3x – 14 = y
x + 2y = 8
2y = -x + 8
2 2 2
y = -1x + 4
2
Write each equation in slope-intercept form.
Identify the slope and y-intercept.
2x + y = 10
-4x + y = 6
4x + 3y = 9
2x + y = 3
5y = 3x
Write the equation of a line in slope-intercept
form that passes through points (3, -4) and
(-1, 4).
1) Find the
slope.
4 – (-4) 8
-1 – 3 -4
m = -2
2) Choose either point and
substitute. Solve for b.
y = mx + b (3, -4)
-4 = (-2)(3) + b
-4 = -6 + b
2=b
Substitute m and b in equation.
Y = mx + b
Y = -2x + 2
Write the equation of the line in
slope-intercept form that passes through
each pair of points.
1)
(-1, -6) and (2, 6)
2)
(0, 5) and (3, 1)
3)
(3, 5) and (6, 6)
4)
(0, -7) and (4, 25)
5)
(-1, 1) and (3, -3)
Graphing an Equation
y = 3x -1
The slope is 3,
use the slope to
plot the second
point

The y-intercept
is -1, so plot
point (0, -1)


Draw a line
through the
two points.
Point-Slope Form
Writing an equation when you know
a point (2, 5) and the slope m = 2
In the graph below, use the information
provided to write the equation of the line.
Use what you know about writing an
equation in slope-intercept form.
Slope = 2 and
point (2,7)
Do you think you can use the same method to
find the y-intercept in the graph below?
Here we must use a different form of writing
an equation and that form is called point-slope.
Slope = 7/3
and point (2,7)
• Suppose you know that a
line passes through the
point (3, 4) with slope 2.
You can quickly write an
equation of the line using
the x- and y-coordinates
of the point and using the
slope.
• The point-slope form of
the equation of a
nonvertical line that
passes through the
(x1, y1) with slope m.
Point-Slope Form
and Writing
Equations
y – y1 = m(x – x1)
(if you know a point and the
slope, use this form)
Let’s try a couple.
Using point-slope form, write the equation of a line
that passes through (4, 1) with slope -2.
y – y1 = m(x – x1)
y – 1 = -2(x – 4)Substitute 4 for x , 1 for y and -2 for m.
1
Write in slope-intercept form.
y – 1 = -2x + 8
y = -2x + 9
1
One last example
Using point-slope form, write the equation of a line
that passes through (-1, 3) with slope 7.
y – y1 = m(x – x1)
y – 3 = 7[x – (-1)]
y – 3 = 7(x + 1)
Write in slope-intercept form
y – 3 = 7x + 7
y = 7x + 10
If you know two points on a line, first use
them to find the slope. Then you can write an
equation using either point.
• Step one – Find the
slope of a line with
points (-4, 3), (-2, 1)
y2  y1
m
x2  x1
1 3
2

 1
 2   4  2
Step Two – Use either point to write the
equation in point-slope form. Use (-4, 3)
y – y1 = m(x – x1)
Y – 3 = -1[x – (-4)]
Y – 3 = -1(x + 4)
Write in slope-intercept form
Y – 3 = -1(x + 4)
Y – 3 = -x - 4
Y = -x - 1
Writing Equations of Parallel
and Perpendicular Lines
Geometry Connection
In coordinate geometry you studied how to
determine if lines where parallel or
perpendicular.
• Nonvertical lines are parallel if they have the same
slope and different y-intercept. Any 2 vertical lines
are parallel. (y = 3x + 1 and y = 3x -3)
• Two lines are perpendicular if the product of their
slopes is -1. A vertical line and a horizontal line are
also perpendicular. (y = -¼x 1 and y = 4x + 2)
In the graph on the left, the two lines are parallel. Parallel
lines have the same slope and never intersect.
In the graph at the right, the two lines are perpendicular.
Perpendicular lines are lines that intersect to form right
angles.
Writing Equations of
Parallel Lines
Write the equation for the line that contains (5, 1) and is
parallel to y = ¼x – 4.
Identify the slope of the given line. y = ¼x – 4 (slope is ¼)
Using point-slope form, write the equation.
y – y1 = m(x – x1)
point-slope form of an equation
y – 1 = ¼(x – 5)
substitute (5, 1) for (x1, y1) and ¼ for m
Y – 1 = ¼x – 5/4
using the Distributive Property, remove the parentheses
Y = ¼x – ¼
simplify and rewrite in slope-intercept
The equation is y = ¼x – 1/4 .
1) Write an equation for the line that contains
(2, -6) and is parallel to y = 2x + 9.
2) Write an equation for the line that
contains (3, 4) and is parallel to y = ½x – 4.
For perpendicular lines, the product of two
numbers is -1, if one number is the negative
reciprocal of the other. Here is how to find
the negative reciprocal of a number.
Start with a fraction: 3
Start with an integer: 4
5
Find its reciprocal: 5
Find its reciprocal:
Write the negative 5

reciprocal:
3
Write the negative 1

reciprocal:
4
3
1
4
Writing Equations of
Perpendicular Lines
Write an equation of the line that contains (0, -2) and is
perpendicular to y = 4x + 3.
Identify the slope of the given line. Y = 4x + 3 (slope is 4, the
negative reciprocal is –¼).
Using point-slope form, write the equation.
y – y1 = m(x – x1)
point-slope form if an equation.
y –(-2) = -¼(x – 0)
substitute (0, -2) for (x1, y1) and –¼ for m
y + 2 = -¼x – 0
using the Distributive Property, remove the parentheses
Y = -¼x - 2
simplify and rewrite in slope-intercept.
The equation is y = -¼x – 2.
1) Write an equation of the line that contains (1, 8)
and is perpendicular to y = ¾x + 1.
2) Write an equation of the line that contains (6, 2)
and is perpendicular to y = -2x + 7
Graphing Absolute
Value Equations
Distance from zero?
A V-shaped graph that points upward or
downward is the graph of an absolute value
equation
• The Absolute value of a
number is its distance from 0
on a number line.
• Make a take of values and
graph the equation y = |x| + 1
Below are the graphs of y = |x| + 1 and y = |x| + 2.
Describe how the graphs are the same and how
they are different.
y = |x| + 1
y = |x| + 2
The graphs are the same shape. The y-intercept of
the first graph is 1. the y-intercept of the second
graph is 2
Describe how each graph below is like y = |x|
and how it is different.
y = |x| + 3
y = |x| - 3
Graph each equation
(function)
1) y = |x| + 2
2) y = |x| – 4
3) y = |x| + 1
4) y = |x| - 5
Use x-values of -2, -1, 0, 1, 2
Equation Forms
(review)
When working with straight lines, there
are often many ways to arrive at an
equation or a graph.
Slope Intercept Form
If you know the slope and where the line crosses the
y-axis, use this form.
y = mx + b
m = slope
b = y-intercept
(where the line crosses the y-axis)
Point Slope Form
If you know a point and the slope, use this form.
y – y1 = m(x – x1)
m = slope
(x1, y1) = a point on the line
Horizontal Lines
y=3
(or any number)
Lines that are horizontal have a slope of zero. They
have “run” but no “rise”. The rise/run formula for
slope always equals zero since rise = o.
y = mx + b
y = 0x + 3
y=3
This equation also describes what is happening to
the y-coordinates on the line. In this case, they
are always 3.
Vertical Lines
x = -2
Lines that are vertical have no slope
(it does not exist).
They have “rise”, but no “run”. The rise/run formula
for slope always has a zero denominator and is
undefined.
These lines are described by what is happening to
their x-coordinates. In this example, the xcoordinates are always equal to -2.
There are several ways to graph a straight line
given its equation.
Let’s quickly refresh our memories on equations of straight
lines:
Slope-intercept
y = mx + b
When stated in “y=”
form, it quickly gives
the slope, m, and
where the line
crosses the y-axis, b,
called the yintercept.
Point-slope
y - y1 = m(x – x1)
when graphing, put
this equation into
“y=” form to easily
read graphing
information.
Horizontal line
Y = 3 (or any #)
Vertical line
X = -2 (or any #)
Horizontal lines have
a slope of zero –
they have “run”, but
no “rise” – all of the
y values are 3.
Vertical line have no
slope (it does not
exist) – they have
“rise”, but no “run” –
all of the x values
are -2.
Remember
If a point lies on a line,
its coordinates make
the equation true.
(2, 1) on the line
y = 2x -3 because
1 = 2(2) - 3
Before graphing a line,
be sure that your
equation starts with
“y =”
To graph 6x + 2y = 8
rewrite the equation:
2y = -6x + 8
Y = -3x + 4
Now graph the line using
either slope intercept
method or table
method.
Practice with Equations
of Lines
Writing and graphing lines
Practice with Equations of Lines
Answer the following questions dealing with equations
and graphs of straight lines.
1)
Which of the following equations passes
through the points (2, 1) and (5, -2)?
a.
c.
y = 3/7x + 5
y = -x + 2
b. y = -x + 3
d. y = -1/3x + 3
2) Does the graph of the straight line
with slope of 2 and y-intercept of 3
pass through the point (5, 13)?
Yes
No
3) The slope of this line is 3/2?
True
False
4) What is the slope of the line
3x + 2y = 12?
a)
b)
c)
d)
3
3/2
-3/2
2
5) Which is the slope of the line
through (-2, 3) and (4, -5)?
a)
b)
c)
d)
-4/3
-3/4
4/3
-1/3
6) What is the slope of the line shown in the
chart below?
X
Y
1
2
3
5
5
8
a)
b)
c)
d)
7
11
1
3/2
3
3/5
7) Does the line 2y + x = 7 pass
through the point (1, 3)?
True
False
8) Which is the equation of a line
whose slope is undefined?
a)
b)
c)
d)
x = -5
y=7
x=y
x+y=0
9) Which is the equation of a line that
passes through (2, 5) and has slope -3?
a)
b)
c)
d)
y = -3x – 3
y = -3x + 17
y = -3x + 11
y = -3x + 5
10) Which of these equations
represents a line parallel to the line
2x + y = 6?
a)
b)
c)
d)
Y = 2x + 3
Y – 2x = 4
2x – y = 8
Y = -2x + 1